You should be guided by your mechanic’s quote for either outcome. In some cases, solenoids are relatively cheap to buy and don’t require too much labour to replace. But, depending on the vehicle, they can also be very expensive and require more or less a full transmission strip-down to remove and replace.

You also need to assess the condition of the whole transmission as putting new parts into a worn gearbox will only delay the inevitable.

This sounds like another one of pesky electrical problems which occur but don't really create havoc beyond making you wonder why it's happening. In many cases, this sort of problem can be caused by a poor earth somewhere on the car or even a chafed wire that is short-circuiting and turning random lights on and off.

But it could also be a symptom of something more serious with the car's electrical systems, so an electronic scan of the vehicle should be your next move. That will rule out a lot of things and also possibly pin-point the problem.

This is indeed a known problem on these transmissions and one for which Mazda issued a Technical Service Bulletin (TSB; like a recall but not safety related) back in 2014. Apparently, a problem with the valve-body (that controls the gearbox’s shifts) caused a scraping or buzzing noise usually when shifting form Park to Drive or Reverse. That’s not exactly what you’re reporting, but it’s almost too similar not to be related. Depending on the build date of the vehicle, Mazda either replaced part of the entire valve-body assembly.

The other problem you’ve alluded to; that of a faulty output speed sensor is a different issue to the one that sounds like it’s causing your problem. In the case of the dud sensor, the gearbox could suddenly decide to shift back to first gear unannounced. If that happened at speed, it could cause the vehicle’s wheels to lock up with some very interesting consequences.